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        <title>Degree (Graph Theory)</title>

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                <h1 id="title" titleSize="">
                    Degree (Graph Theory)
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            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p>Let $G$ be a (simple) <a href=https://zhaoshenzhai.github.io/mathwiki/graph.md class="internalLink references" title="Graph" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/graph.md&#34;, {&#34;Date&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/graph.md&#34;, {&#34;Date&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-05T18:13:24-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">graph</a>. Although very crude, good control over the <em>degree</em> of a vertex in $G$ can form the basis of some interesting combinatorial results when $G$ is <a href=https://zhaoshenzhai.github.io/mathwiki/finite_graph.md class="internalLink references" title="Finite Graph" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/finite_graph.md&#34;, {&#34;Date&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Finite Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/finite_graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Finite Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/finite_graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/finite_graph.md&#34;, {&#34;Date&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-06T23:44:59-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Finite Graph&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/finite_graph&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">finite</a>. If $G$ is <a href=https://zhaoshenzhai.github.io/mathwiki/infinite_graph.md class="internalLink references ghostLink" title="infinite" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/infinite_graph.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/infinite_graph.md&#34;, &#34;nopPage&#34;);">infinite</a>, this notion can instead be used as a measure of the local complexity of $G$.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>The <em>degree</em> of a vertex $v\in V(G)$ is the <a href=https://zhaoshenzhai.github.io/mathwiki/cardinal class="internalLink references ghostLink" title="cardinality" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/cardinal&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/cardinal&#34;, &#34;nopPage&#34;);">cardinality</a> $d_G(v)$ of its <em>neighbors</em> $N_G(v)\coloneqq\l\{u\in V\st uv\in E\r\}$.</div>

<h1 id="degrees-in-finite-graphs">Degrees in Finite Graphs</h1>
<h2 id="handshaking_lemma">Handshaking Lemma</h2>
<p>We have $\sum_{v\in V}d_G(v)=2|E(G)|$, which is intuitively obvious since summing all the degrees counts each edge twice, once from each of its ends. Formally, we consider the map $i_v:E\to2$ with $e\mapsto1$ iff $v\in e$, and observe that
$$\begin{equation}
\sum_{v\in V}d_G(v)=\sum_{v\in V}\sum_{e\in E}i_v(e)=\sum_{e\in E}\sum_{v\in V}i_v(e)=\sum_{e\in E}2=2|E(G)|.
\end{equation}$$
An immediate corollary of this result is the so-called <em>Handshaking Lemma</em>, stating that the number of vertices of odd degree in a finite graph $G$ is always even. This <em>fails</em> for infinite graphs: consider, for instance, the infinite ray $\N$.</p>
<blockquote>
<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect">Proof.</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent">Splitting off the even-degree vertices and keeping only the odd-degrees ones $v_i,\dots,v_m$, we see that $\sum_{i\leq m}(2n_i+1)$ is even, where $2n_i+1=d_G(v_i)$. Hence $m+2\sum_{i\leq m}n_i$ is even, and thus so is $m$.<span style="float:right;">$\blacksquare$</span></span></div>

</blockquote>
<p>If $G$ is finite, the <em>average</em> degree $d(G)\coloneqq\frac{1}{|V(G)|}\sum_{v\in V}d_G(v)$ of $G$ is, by the result above, twice the ratio $\epsilon(G)\coloneqq|E(G)|/|V(G)|$.</p>
<h2 id="large-degree-graphs">Large-degree Graphs</h2>
<p><span style="color:red"><strong>TODO</strong> (Dirac’s theorem).</span></p>
<h1 id="locally-finite-graphs">Locally-finite Graphs</h1>


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                September 6, 2024 | Zhaoshen Zhai

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